On Complex and Noncommutative Tori

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 4, Pages 973–981 S 0002-9939(05)08244-4 Article electronically published on September 28, 2005

ON COMPLEX AND NONCOMMUTATIVE TORI

IGOR NIKOLAEV

(Communicated by Michael Stillman)

Abstract. The “noncommutative geometry” of complex algebraic curves is studied. As a ﬁrst step, we clarify a morphism between elliptic curves, or ∗ 2πiθ complex tori, and C -algebras Tθ = {u, v | vu = e uv},ornoncommutative tori. The main result says that under the morphism, isomorphic elliptic curves map to the Morita equivalent noncommutative tori. Our approach is based on the rigidity of the length spectra of Riemann surfaces.

Introduction Noncommutative geometry is a branch of algebraic geometry studying “varieties” over noncommutative rings. The noncommutative rings are usually taken to be rings of operators acting on a Hilbert space [7]. The rudiments of noncommutative geometry can be traced back to F. Klein [3], [4] or even earlier. The fundamental modern treatise [1] gives an account of status and perspective of the subject. ∗ The noncommutative torus Tθ is a C -algebra generated by linear operators u and v on the Hilbert space L2(S1) subject to the commutation relation vu = e2πiθuv, θ ∈ R − Q [11]. The classiﬁcation of noncommutative tori was given in [2], [8], [11]. Recall that two such tori Tθ,Tθ are Morita equivalent if and only if θ, θ lie in the same orbit of the action of group GL(2, Z) on irrational numbers by linear fractional transformations. It is remarkable that the “moduli problem” for Tθ looks as such for the complex tori Eτ = C/(Z + τZ), where τ is complex modulus. Namely, complex tori Eτ ,Eτ are isomorphic if and only if τ,τ lie in the same orbit of the action of SL(2, Z) on complex numbers by linear fractional transformations. It was observed by some authors (e.g. [5], [15]) that it might not be just a coincidence. This note is an attempt to show that it is indeed so: there exists a general morphism between Riemann surfaces and C∗-algebras. Let us give rough idea of our approach. Given Riemann surface S,thereisa → R∞ function S + which maps the (discrete) set of closed geodesics of S to a discrete subset of a real line by assigning each closed geodesic its riemannian length. If Tg(S) is the space of all Riemann surfaces of genus g ≥ 0, then the function −→ R∞ (1) W : Tg(S) +

Received by the editors February 25, 2003 and, in revised form, November 2, 2004. 2000 Mathematics Subject Classiﬁcation. Primary 14H52, 46L85. Key words and phrases. Elliptic curve, noncommutative torus.

c 2005 American Mathematical Society 973

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is ﬁnite-to-one and “generically” one-to-one [17]. In the case g = 1, function W is always one-to-one. It is known also that restriction Wsyst : Tg(S) → R+ of W to 0 the shortest closed geodesic of S (called systole)isaC Morse function on Tg(S) [13], §5. Below we focus on the case g = 1, i.e. T1 Eτ . Recall that Tθ has a unique state s0 (which is actually a tracial state) [11]. Any positive functional on Tθ has form ωs0,whereω>0 is a real number. Let Θ={Tθ | θ ∈ R − Q} and Ω = {ω ∈ R | ω>0}. We deﬁne a map × −→ R∞ (2) V :Θ Ω + by the formula (T ,ω) →{f (ω)lntr(A )}∞ ,where θ n n n=0 a0 1 A0 = , 10 a0 1 a1 1 A1 = 10, 10, . (3) . a0 1 a1 1 an 1 An = 10 10... 10

are integer matrices whose entries ai > 0 are partial denominators of the continued 0 fraction expansion of θ,andfn are monotone C functions of ω. Assuming that −1 functions W, V have common range, one gets a mapping WV : Eτ → (Tθ,ω). Morphisms between Eτ and Tθ have been studied in [5], [9], [10], [16]. The works [5], [10] and [16] treat noncommutative tori as “quantum compactiﬁcation” of the space of elliptic curves. This approach deals with an algebraic side of the subject. In particular, Manin [5] suggested to use “pseudolattices” (i.e. K0-group of Tθ)to solve the multiplication problem for real number ﬁelds. This problem is part of the Hilbert 12th problem. In [9] a functor from derived category of holomorphic vector bundles over Tθ to the Fukaya category of such bundles over Eτ was constructed. In this note we prove the following results.

Theorem 1. Let Eτ be a complex torus of modulus τ,Im τ > 0,andlet(Tθ,ω) be a pair consisting of noncommutative torus with an irrational Rieﬀel’s parameter θ and a positive functional Tθ → C of norm ω. Then there exists a one-to-one mapping Eτ → (Tθ,ω). The action of the modular group SL(2, Z) on the complex half plane {τ ∈ C | Im τ > 0} is equivariant with: (i) the action of group GL(2, Z) on irrationals {θ ∈ R − Q | θ>0} by linear fractional transformations; (ii) a discrete action on positive reals {ω ∈ R | ω>0}. In particular, isomorphic complex tori map to the Morita equivalent noncommutative tori, and vice versa.

Deﬁnition 1. The irrational number θ of mapping Eτ → (Tθ,ω) we call a projective curvature of the elliptic curve Eτ . Theorem 2. Projective curvature of an elliptic curve with complex multiplication is a quadratic irrationality.

1. Proofs The proof of both theorems is based on the rigidity of length spectrum of complex torus; cf. Wolpert [17]. Preliminary information on complex and noncommutative tori can be found in Section 2.

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1.1. Proof of Theorem 1. Let us review the main steps of the proof. By the rigidity lemma (Lemma 1) the length spectrum Sp E defines conformal structure of E. In fact, this correspondence is a bijection. Under isomorphisms of E the length spectrum can acquire a real multiple or get a “cut of finite tail” (Lemma 2). We attach to C/L a continued fraction of its projective curvature θ as specified in the Introduction. Then isomorphic tori C/L will have continued fractions which differ only in a finite number of terms. In other words, one can attach a Morita equivalence class of noncommutative tori to every isomorphism class of complex tori. Lemma 1 (Rigidity of length spectrum). Let Sp E be the length spectrum of a complex torus E = C/L. Then there exists a unique complex torus with the spectrum. This correspondence is a bijection. Proof. See McKean [6].

Let Sp X = {l1,l2,...} be the length spectrum of a Riemann surface X.Leta> 0bearealnumber.ByaSp X we understand the length spectrum {al1,al2,...}. Similarly, for any m ∈ N we denote by Spm X the length spectrum {lm,lm+1,...}, i.e. the one obtained by deleting the ﬁrst (m − 1)-geodesics in Sp X. Lemma 2. Let E ∼ E be isomorphic complex tori. Then either: (i) Sp E = |α|Sp E for an α ∈ C×,or (ii) Sp E = Spm E for a m ∈ N. Proof. (i) The complex tori E = C/L, E = C/M are isomorphic if and only if M = αL for a complex number α ∈ C×. It is not hard to see that the closed geodesic of E are bijective with the points of the lattice L = ω1Z + ω2Z in the following way. Take a segment of a straight line through points 0 and ω of lattice L which contains no other points of L. This segment represents a homotopy class of curves through 0 and a closed geodesic of E. Evidently, this geodesic will be the shortest in its homotopy class with the length |ω| equal to an absolute value of the complex number ω.Thus,|ω| belongs to the length spectrum of E. Now let Sp E = {|ω1|, |ω2|,...} with ωi ∈ L.SinceM = αL,onegetsSp E = {|α||ω1|, |α||ω2|,...} and Sp E = |α|Sp E. Item (i) follows. (ii) Note that according to (i) the length spectrum Sp X = {l0,l1,l2,...} can be written as Sp X = {1,l1,l2,...} after multiplication on 1/l0,wherel0 is the length of the shortest geodesic. Note also that the shortest geodesic of the complex torus has a homotopy type (1, 0) or (0, 1) (standard generators for π1E). Let a, b, c, d be integers such that ad − bc = ±1andlet ω1 = aω1 + bω2, (4) ω2 = cω1 + dω2

be an automorphism of the lattice L = ω1Z + ω2Z. This automorphism maps standard generators (1, 0) and (0, 1) of L to the vectors ω1 =(a, b),ω2 =(c, d). Let their lengths be lm,lm+1, respectively. As we showed earlier, lm,lm+1 ∈ Sp E, and it is not hard to see that there are no geodesics of the intermediate length. (This gives a justiﬁcation for the notation chosen.) Note that ω1,ω2 are standard generators for the complex torus E ∼ E, and therefore one of them is the shortest closed geodesics of E. One can normalize it to the length 1.

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On the other hand, there are only a finite number of closed geodesics of length smaller than ln (McKean [6]). Thus Sp E ∩ Sp E = {lm,lm+1,...} for a finite number m and since (4) is automorphism of the lattice L.Inotherwords,Sp E = Spm E. Item (ii) follows. To finish the proof of item (i) of Theorem 1, one needs to combine Lemmas 1 and 2 with the fact that two noncommutative tori Tθ,Tθ are Morita equivalent if and only if their continued fractions differ only in a finite number of terms (Section 2.1). To prove item (ii) of Theorem 1, let to the contrary the action of SL(2, Z)be

nondiscrete, i.e. having limit points in Ω. Let p = limn→∞(Tθn ,ωn), where θn lie inthesameorbitofGL(2, Z). Let Ep be the corresponding complex torus such ∼ that Ep = Epn are nonisomorphic. By continuity of the systole function Wsyst (see

the Introduction), Sp Ep = lim Sp Epn . Then by the rigidity of length spectra, ∼ Ep = Epn are isomorphic. The contradiction proves item (ii) of the theorem. 1.2. ProofofTheorem2. Let us outline the idea of the proof. If E admits complex multiplication, then its complex modulus τ lies in an imaginary quadratic field K. In fact, up to an isogeny, the ring of endomorphisms End E = OK ,whereOK is the ring of integers of field K.ItcanbeshownthatL is an ideal in OK (Section 2.3). The length spectrum Sp E of an elliptic curve with complex multiplication is a “geometric progression” with the growth rate |α|,whereα ∈ End E (Lemma 3). One can use Klein’s lemma (Lemma 4) to characterize length spectra in terms of continued fractions. In particular, length spectrum with asymptotically geometric growth correspond to periodic continued fractions. Thus, projective curvature converges to quadratic irrationality. Definition 2. Length spectrum Sp E of an elliptic curve E = C/L is called α- multiplicative, if there exists a complex number α ∈ C× with |α| > 1 such that n n (5) Sp E = {l1,...,lN , |α|l1,...,|α|lN ,...,|α| l1,...,|α| lN ,...}, for N ∈ N. Lemma 3. Let E be an elliptic curve with complex multiplication. Then its length spectrum Sp E is α-multiplicative for an α ∈ C×.

Proof. Let E = C/L be√ a complex torus which admits nontrivial endomorphisms z → αz, α ∈ K = Q( −d). It is known that End E is an order in the field K. In fact, up to an isogeny of E, End E OK ,whereOK is the ring of integers of imaginary quadratic field K (Section 2.3). Lattice L in this case corresponds to an ideal in OK . Let l1 be the minimal length of a closed geodesic of E. For an endomorphism × α : E → E,α ∈ C , consider the set of geodesics whose lengths are less than |α|l1. By the properties of Sp E mentioned in Section 1.1, such a set will be finite. Let us denote the lengths of geodesics in this set by l1,...,lN . Since every geodesic in Sp E is a complex number ωi lying in the ring L ⊆OK , one can consider the set of geodesics αω1,...,αωN . The length of these geodesics will be |α|l1,...,|α|lN , respectively. It is not hard to see that by the choice of number N, the first 2N elements of Sp E are presented by the following growing sequence of geodesics: l1,...,lN , |α|l1,...,|α|lN . We proceed by iterations of α,untilallclosedgeodesics of E are exhausted. The conclusion of Lemma 3 follows.

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We shall need the following statement regarding geometry of the regular continued fractions [3], [4]. It is valid for any regular fraction, not necessarily periodic. Lemma 4 (F. Klein). Let 1 (6) ω = µ + 1 1 µ2 + µ3 + ... be a regular continued fraction. Let us denote the convergents of ω by p− 0 p 1 p µ p µ p − + p − (7) 1 = , 0 = , 1 = 1 ,..., ν = ν ν 1 ν 2 . q−1 1 q0 0 q1 1 qν µν qν−1 + qν−2

For any lattice L in C, consider a segment I with ends in the points (pν−2,qν−2) and (pν ,qν ). Then the segment J which joins 0 with the point pν−1,qν−1 is parallel to I and

(8) |I| = µν |J|, where |•| denotes the length of the segment. Proof. We refer the reader to [4].

Corollary 1. Let ων =(pν ,qν ) be lattice points mentioned in Lemma 4.Then the length of vector ων can be evaluated with the help of the following asymptotic formula:

(9) |ων |≈|ων−2| + µν |ων−1|.

Proof. Indeed, using the notation of Lemma 4, one can write |(pν ,qν )|≈ |(pν−2,qν−2)|+|I|. But according to equation (8), |I| = µν |(pν−1,qν−1)|. Corollary 1 follows.

Note that according to the recurrent formula (9) the length spectrum {|ων |} coming from continued fraction (6) is completely determined by the ﬁrst two values: |ω1| and |ω2|. Using (9), one can easily deduce the following asymptotic formula for |ων | as a function of |ω1|, |ω2|: ν ν (10) |ων |≈|ω2| µk + |ω1| µk + O(ν). k=3 k=4 Fix N a positive integer. It follows from equation (10) that |ω | µ ...µ |ω | + µ ...µ |ω | + O(ν) lim ν+N = µ ...µ lim ν 3 2 ν 4 1 →∞ ν+1 ν+N →∞ ν |ων | ν µν ...µ3|ω2| + µν ...µ4|ω1| + O(ν)

(11) = µν+1 ...µν+N . Let E be an elliptic curve with complex multiplication. Then by Lemma 3 its length spectrum Sp E is α-multiplicative. In other words, l (12) ν+N = |α| = Const, lν for an N ∈ N and any νmodN.Notethat|α| is a rational integer. Thus, by formula (11) we have µν+1 ...µν+N = Const, for any νmodN.Thelast requirement can be satisﬁed if and only if continued fraction (6) is N-periodic. Theorem 2 is proven.

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2. Background information In the present section we brieﬂy review noncommutative and complex tori. The excellent source of information on noncommutative torus are papers [2], [11] and a monograph of [12]. The literature on complex torus is fairly vast. We recommend for the reference Ch. VI of [14].

2.1. Noncommutative torus. By the C∗-algebra one understands a noncommutative Banach algebra with an involution [12]. Namely, a C∗-algebra A is an algebra over C with a norm a →||a|| and an involution a → a∗,a∈ A, such that A is complete with respect to the norm, and such that ||ab|| ≤ ||a|| ||b|| and ||a∗a|| = ||a||2 for every a, b ∈ A.IfA is commutative, then the Gelfand theorem says that A is ∗ isometrically ∗-isomorphic to the C -algebra C0(X) of continuous complex-valued functions on a locally compact Hausdorff space X. For otherwise, A represents a “noncommutative” topological space X. ∗ ∗ K0 and dimension groups. Given a C -algebra, A, consider a new C -algebra Mn(A), i.e. the matrix algebra over A. There exists a remarkable semi-group, + ∞ A , connected to the set of projections in algebra M∞ = n=1 Mn(A). Namely, projections p, q ∈ M∞(A) are Murray-von Neumann equivalent p ∼ q if they can ∗ ∗ be presented as p = v v and q = vv for an element v ∈ M∞(A). The equivalence class of projections is denoted by [p]. The semi-group A+ is defined to be the set of all equivalence classes of projections in M∞(A) with the binary operation [p]+[q]=[p ⊕ q]. The Grothendieck completion of A+ to an abelian group is ∗ called a K0-group of A. The functor A → K0(A) maps the unital C -algebras into the category of abelian groups so that the semi-group A+ ⊂ A corresponds + ⊂ ∈ toa“positivecone”K0 K0(A) and the unit element 1 A corresponds to the ∈ + “order unit” [1] K0(A). The ordered abelian group (K0,K0 , [1]) with the order unit is called a dimension (Elliott) group of A. The dimension (Elliott) group is a complete invariant of the AF C∗-algebras. Noncommutative torus. Fix θ irrational and consider a linear flowx ˙ = θ, y˙ =1on the torus. Let S1 be a closed transversal to our flow. The noncommutative torus ∗ Tθ is a norm-closed C -algebra generated by the unitary operators in the Hilbert space L2(S1): Uf(t)=z(t)f(t),Vf(t)=f(t − α),

which are multiplication by a unimodular function z(t) and rotation operators. It 2πiα could be easily verified that UV = e VU. As an “abstract” algebra, Tθ is a ∗ 1 ∗ crossed product C -algebra C(S ) φ Z of a (commutative) C -algebra of complex- valued continuous functions on S1 by the action of powers of φ,whereφ is a rotation 1 of S through the angle 2πα. Tθ is not AF , but can be embedded into an AF - algebra whose dimension group is Pθ (to be specified below); the latter is known to be intimately connected with the arithmetic of the irrational numbers θ’s. The following beautiful result is due to the efforts of many mathematicians1 (Effros, Elliott, Pimsner, Rieffel, Shen, Voiculescu, etc.).

1The author apologizes for possible erroneous credits regarding the history of the problem. Classiﬁcation of noncommutative tori seems to be an old problem; early results in this direction can be found in the works of Klein [3], [4].

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Theorem 3 (Classiﬁcation of noncommutative tori). Let Tθ be a noncommutative torus. Suppose that the θ has a continued fraction expansion 1 θ = a + def=[a ,a ,a ,...]. 0 1 0 1 2 a + 1 a + ... 2 Let ϕ be a composition of isometries of the lattice Z2 ⊂ R2 : ϕ = a0 1 n n 10 an 1 ... 10.ThenTθ can be embedded into an AF -algebra whose dimension group 2 is a direct limit of the ordered abelian groups: Pθ = limn→∞(Z ,ϕn). Moreover, if θ =[a0,a1,...] and θ =[b0,b1,...] are two irrational numbers, then Pθ and Pθ are isomorphic (i.e. noncommutative tori Tθ and Tθ are Morita equivalent) if and only if am+k = bm for an integer number k ∈ Z. In other words, the irrational numbers aθ+b − ± ∈ Z θ and θ are modular equivalent: θ = cθ+d ,adbc = 1,wherea, b, c, d are integer numbers. Proof. An algebraic proof of this fact can be found in [2]. 2.2. Complex torus. Let L denote a lattice in the complex plane C. Attached to L, there are the following classic Weierstrass function ℘(z; L) and Eisenstein series G (L): k 1 1 1 (13) ℘(z; L)= + − , z2 (z + ω)2 ω2 ω∈L× k≥2 1 (14) G (L)= . k ω2k ω∈L×

℘(z; L)isanalytic,andGk(L) is convergent for any lattice L [14]. There exists a duality between lattices L and cubic curves E given by the following theorem. C → 1 Theorem 4. Let L be a lattice in . Then the map z (℘(z; L), 2 ℘ (z; L)) is an analytic isomorphism from complex torus C/L to elliptic cubic E = E(C): 2 2 3 (15) E(C)={(x, y) ∈ C | y = x − 15G4(L)x − 35G6(L)}. Conversely, to any cubic in the Weierstrass normal form y2 = x3 + ax + b there corresponds a unique lattice L such that a = −15G4(L) and b = −35G6(L). Proof. We refer the reader to [14] for a detailed proof of this fact. Isomorphism of complex tori. Let L be a lattice in C. The Riemann surface C/L is called a complex torus.Letf : C/L → C/M be a holomorphic and invertible map (isomorphism) between two complex tori. Since f is covered by a linear map z → αz on C, one can easily conclude that αL = M for an α ∈ C×. On the other × hand, lattice L can always be written as L = ω1Z + ω2Z,whereω1,ω2 ∈ C and ω2 ω2 = kω1 for a k ∈ R. The complex number τ = is called a complex modulus of ω1 lattice L. Lemma 5. Two complex tori are isomorphic if and only if their complex moduli τ and τ satisfy the equation aτ + b (16) τ = ad − bc = ±1, a,b,c,d∈ Z. cτ + d Proof. The proof of this fact can be found in [14].

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2.3. Elliptic curves with complex multiplication. Let E = C/L be an elliptic curve. Consider the set End E of analytic self-mappings of E.Eachf ∈ End E is covered on the complex plane by map z → αz for an α ∈ C.Itisnothard to see that End E has the structure of a ring under the pointwise addition and multiplication of functions. The set End E is called an endomorphism ring of an elliptic curve E. By the remarks above, End E can be thought of as a subring of complex numbers: (17) End E = {α ∈ C | αL ⊂ L}. There exists a fairly complete algebraic description of such rings. Roughly speaking, they are either “rational integers” Z or integers OK of an algebraic number field K. The following lemma is true. Lemma 6. Let α ∈ End E be a complex number. Then either: (i) α is a rational integer, or √ (ii) α is an algebraic integer in an imaginary quadratic number field K =Q( −d). Proof. See [14]. Complex multiplication. If End E is different from Z, E is said to be an elliptic curve with complex multiplication.IfE admits complex multiplication, then its ring End E is an order in an imaginary quadratic field K.Infact,E admits an isogeny (analytic homomorphism) to a curve E such that End E OK ,whereOK is the ring of integers of field K [14]. Thus, by property αL ⊆ L, lattice L is an ideal in OK .DenotebyhK the class number of field K. It is well known that there exist O C hK nonisomorphic ideals in K . Therefore, elliptic curves E1 = /L1,...,EhK = C /LhK are pairwise nonisomorphic, but their endomorphism ring is the same [14]. Acknowledgements It is my pleasure to thank G. A. Elliott, Yu. I. Manin and M. Rieffel for their interest in the subject of present note. I am grateful to the referee for his critical remarks and helpful suggestions. References

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Department of Mathematics, University of Calgary, 2500 University Drive N.W., Calgary, Canada T2N 1N4 E-mail address: [email protected]

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